"Kyriarchy" means what "patriarchy" originally meant. "Patr-" doesn't mean man (that would be "vir"), it's Latin for "father"; in Roman society, a paterfamilias was a head of household, which included junior men and slaves. The paterfamilias social organization was the ancestor of Western feudalism and the concept of the "lord" and the great chain of being. The whole point of the concept of patriarchy is that it was precisely "a complex pyramidal system of intersecting multiplicative social structures of superordination and subordination, of ruling and oppression."

Here's hoping rebranding it sells a few more bottles, but I think you're right that putting a pretty name on a ugly thing is not wise.